Low bias estimation of small signal-to-noise ratio

ABSTRACT

A method implemented by a user equipment includes selecting a first estimate of a signal-to-noise (SNR) ratio, calculating a first amplitude and first noise variance, calculating a second amplitude and a second noise variance, calculating a second SNR, calculating a resolution value, adjusting the first SNR, and performing estimation iterations until the resolution value is equal to a predetermined value.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.12/060,541, filed Apr. 1, 2008, which is a continuation of U.S. patentapplication Ser. No. 10/867,508, filed Jun. 14, 2004 and issued on May20, 2008 as U.S. Pat. No. 7,376,178, which is a continuation of U.S.patent application Ser. No. 10/269,606, filed Oct. 11, 2002 and issuedon Jul. 6, 2004 as U.S. Pat. No. 6,760,370, which claims the benefit ofU.S. Provisional Patent Application Ser. No. 60/369,655 filed Apr. 3,2002, which are incorporated by reference as if fully set forth.

FIELD OF THE INVENTION

The invention relates to processing of communications signals. Moreparticularly, the invention relates to the estimation of thecommunication signal power in terms of signal-to-noise ratio.

BACKGROUND OF THE INVENTION

In the field of communications, various types of systems use algorithmsthat depend on a signal-to-noise ratio (SNR) estimate for properoperation. Code division multiple access (CDMA) systems, such as timedivision duplex CDMA (TDD/CDMA) and time division synchronous CDMA(TDSCDMA) and frequency division duplex CDMA (FDD/CDMA) and CDMA 2000,use SNR estimation for power control to maintain the required linkquality while using the minimum transmitted power. An asymmetric digitalsubscriber loop (ADSL) system uses SNR for the bit allocation algorithmto select the maximum transmission data rate. In turbo decoders, boththe determined signal power and noise power are required. Rate adaptivetransmission systems often use SNR to dynamically adapt the modulationscheme or the coding rate.

Several algorithms are known for performing SNR estimation. One suchalgorithm, the received data-aided (RXDA) estimation, is based on thefollowing equation:

$\begin{matrix}{{SNR} = \frac{\left( {\frac{1}{N}{\sum\limits_{k = 1}^{N}{r_{k}}}} \right)^{2}}{{\frac{1}{N}{\sum\limits_{k = 1}^{N}r_{k}^{2}}} - \left( {\frac{1}{N}{\sum\limits_{k = 1}^{N}{r_{k}}}} \right)^{2}}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

where r is the received signal vector and N is the number of samplepoints read by the receiver for the vector r.

Another known algorithm is the transmitted data-aided (TXDA) algorithm,which is represented by the equation:

$\begin{matrix}{{SNR} = \frac{\left( {\frac{1}{N}{\sum\limits_{k = 1}^{N}{r_{k}a_{k}}}} \right)^{2}}{{\frac{1}{N - 3}{\sum\limits_{k = 1}^{N}r_{k}^{2}}} - {\frac{1}{N\left( {N - 3} \right)}\left( {\sum\limits_{k = 1}^{N}{r_{k}a_{k}}} \right)^{2}}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

A third known algorithm for SNR estimation is represented as:

$\begin{matrix}{{SNR} = {\frac{N}{2}\left( {\sum\limits_{k = 1}^{N/2}\frac{\left( {{r_{{2\; k} - 1}} - {r_{2\; k}}} \right)^{2}}{r_{{2\; k} - 1}^{2} + r_{2\; k}^{2}}} \right)^{- 1}}} & {{Equation}\mspace{14mu} 3}\end{matrix}$

The algorithms for Equations 1 and 3 are performed blind without anypilot signal. In contrast, the TDXA algorithm uses a pilot signal withknown training sequences, which provides enhanced performance. Thedrawback of TDXA is that additional equipment is required to process thetraining sequence. Although the RXDA and Equation 3 algorithms work wellwhen the SNR is high, their performance suffers at low and negativeSNRs, where they are known to have a high bias. This is problematic forvarious communication systems. For example, turbo code applications areknown to experience negative ratios of symbol energy to noise density.In CDMA systems, the chip energy to noise density is often negative.Hence, there is a need to develop a blind SNR estimation method thatworks well at low and negative values without the benefit of a trainingsequence.

SUMMARY

An apparatus and method for low bias estimation of small or negativesignal-to-noise ratio (SNR) for a communication signal is presented. Theestimation is iterative and comprises choosing an initial minimum andmaximum estimate of the signal amplitude and determining the meanthereof. Associated minimum and maximum noise variances are calculatedbased on the amplitude values. Using probability density, maximumlikelihood estimates of the minimum, maximum and mean amplitudes arederived. Based on whether the mean amplitude estimate increases ordecreases, the initial minimum or maximum estimate is set equal to themaximum likelihood mean amplitude, and the resolution between the newminimum and maximum estimates is determined. These steps are repeateduntil the resolution is within the acceptable limit, at which point theSNR is calculated from the mean amplitude.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a process flow diagram of method 100 for SNR estimation.

FIG. 2 shows a process flow diagram of method 200 for SNR estimation.

FIG. 3 shows a graph of calculated SNRs plotted against assumed SNRsusing method 100.

FIG. 4 shows a comparison of mean SNR estimates of a BPSK signalperformed by method 200, RXDA, TXDA and a third other algorithm, withsampling N=1024.

FIG. 5 shows a comparison of mean square errors (MSE) normalized to SNRof a BPSK signal, performed by method 200, RXDA, TXDA, a third algorithmand the Cramer-Rao (CR) bound, with sampling N=1024.

FIG. 6 shows a comparison of mean SNR estimates of an 8PSK signalperformed by method 200, a decision directed algorithm, RXDA and TXDA,with sampling N=100.

FIG. 7 shows a comparison of mean square errors (MSE) normalized to SNRof an 8PSK signal performed by method 200, a decision directedalgorithm, RXDA, TXDA and the Cramer-Rao (CR) bound, with samplingN=100.

FIG. 8 shows a comparison of mean SNR estimates of an 8PSK signalperformed by method 200, a decision directed algorithm, RXDA and TXDA,with sampling N=1024.

FIG. 9 shows a comparison of mean square errors (MSE) normalized to SNRof an 8PSK signal performed by method 200, a decision directedalgorithm, RXDA, TXDA and the Cramer-Rao (CR) bound, with samplingN=1024.

FIG. 10 shows a comparison of mean SNR estimates of a 16PSK signalperformed by method 200, a decision directed algorithm, RXDA and TXDA,with sampling N=100.

FIG. 11 shows a comparison of mean square errors (MSE) normalized to SNRof a 16PSK signal performed by method 200, a decision directedalgorithm, RXDA, TXDA and the Cramer-Rao (CR) bound, with samplingN=100.

FIG. 12 shows a comparison of mean SNR estimates of a 16PSK signalperformed by method 200, a decision directed algorithm, RXDA and TXDA,with sampling N=1024.

FIG. 13 shows a comparison of mean square errors (MSE) normalized to SNRof a 16PSK signal performed by method 200, a decision directedalgorithm, RXDA, TXDA and the Cramer-Rao (CR) bound, with samplingN=1024.

FIG. 14 shows a convergence of estimation iterations for several trialsof method 200.

FIG. 15 shows a system for communications between a base station anduser equipments employing SNR estimation methods 100 and 200.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

For a BPSK modulated signal, the time and carrier phase synchronizationcan be obtained so the received samples can be expressed as:

r _(k) =s _(k) +n _(k),   Equation 4

where s_(k) is the transmitted signal taking amplitude values from {−A,A} with equal probability and n_(k) is real additive white Gaussiannoise with variance of σ². In order to determine the unknown value A, aprobability density function is a preferred technique. The probabilitydensity function of r_(k) can be expressed as:

$\begin{matrix}{{f\left( r_{k} \right)} = {\frac{1}{2}\left\{ {{f_{+}\left( r_{k} \right)} + {f_{-}\left( r_{k} \right)}} \right\}}} & {{Equation}\mspace{14mu} 5}\end{matrix}$

where

$\begin{matrix}{{f_{+}\left( r_{k} \right)} = {\frac{1}{\sqrt{2\; \pi}\sigma}^{- \frac{{({r_{k} - A})}^{2}}{2\; \sigma^{2}}}}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

and

$\begin{matrix}{{f_{-}\left( r_{k} \right)} = {\frac{1}{\sqrt{2\; \pi}\sigma}{^{- \frac{{({r_{k} + A})}^{2}}{2\; \sigma^{2}}}.}}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

For a received sample of consecutive symbols of length N (r₁,r₂, . . .,r_(N)), the probability density function can be expressed as:

$\begin{matrix}{{f_{N}\left( {r_{1},r_{2},\ldots \mspace{14mu},r_{N}} \right)} = {\prod\limits_{k = 1}^{N}\; {f\left( r_{k} \right)}}} & {{Equation}\mspace{14mu} 8}\end{matrix}$

An equation for amplitude A which maximizes the probability function canbe determined by taking the partial derivative of Equation 8 withrespect to amplitude A, and setting the partial derivative equal tozero:

$\begin{matrix}{\frac{\partial{f_{N}\left( {r_{1},r_{2},\ldots \mspace{14mu},r_{N}} \right)}}{\partial A} = 0} & {{Equation}\mspace{14mu} 9}\end{matrix}$

The determination of a maximum likelihood estimate of A is then thesolution to Equation 10:

$\begin{matrix}{A = {\frac{1}{N}{\sum\limits_{k = 1}^{N}{r_{k}{{th}\left( \frac{{Ar}_{k}}{\sigma^{2}} \right)}}}}} & {{Equation}\mspace{14mu} 10}\end{matrix}$

where

$\begin{matrix}{{{th}(x)} = {\frac{^{x} - ^{- x}}{_{x} + ^{- x}}.}} & {{Equation}\mspace{14mu} 11}\end{matrix}$

Since the SNR is unknown, it may possibly be high or low. If the SNR ishigh, an acceptable approximation for value th can be made as follows:

$\begin{matrix}{{{th}\left( \frac{{Ar}_{k}}{\sigma^{2}} \right)} \cong \left\{ \begin{matrix}{{+ 1},{r_{k} > 0}} \\{{- 1},{r_{k} < 0}}\end{matrix} \right.} & {{Equation}\mspace{14mu} 12}\end{matrix}$

The decision-directed amplitude estimate is then:

$\begin{matrix}{\hat{A} = {\frac{1}{N}{\sum\limits_{k = 1}^{N}\; {r_{k}}}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$

The noise power can be estimated as total power minus the signal power,and the SNR can therefore be estimated as:

$\begin{matrix}{{S\; N\; R} = \frac{\left( {\frac{1}{N}{\sum\limits_{k = 1}^{N}\; {r_{k}}}} \right)^{2}}{{\frac{1}{N}{\sum\limits_{k = 1}^{N}r_{k}^{2}}} - \left( {\frac{1}{N}{\sum\limits_{k = 1}^{N}\; {r_{k}}}} \right)^{2}}} & {{Equation}\mspace{14mu} 14}\end{matrix}$

In an alternative embodiment, for a signal in which the timesynchronization and the carrier phase synchronization have been obtainedfor MPSK modulation, the value s_(k) of Equation 4 is the transmittedM-ary PSK signal, represented as:

Ae ^(j2πk/M) ,k=0,1, . . . ,M−1   Equation 15

with equal probability of 1/M, and A as the amplitude of MPSK signals_(k). Value n_(k) from Equation 4 is the complex additive whiteGaussian noise with variance of 2σ². The probability density function ofr_(k), where

r _(k) =x _(k) +jy _(k)   Equation 16

can be expressed as:

$\begin{matrix}{{f\left( {x_{k},y_{k}} \right)} = {\frac{1}{M}{\sum\limits_{l = 0}^{M - 1}\; {\frac{1}{\sqrt{2\; \pi}\sigma}\exp \left\{ {- \frac{\begin{matrix}{\left( {x_{k} - {X_{l}A}} \right)^{2} +} \\\left( {y_{k} - {Y_{l}A}} \right)^{2}\end{matrix}}{2\; \sigma^{2}}} \right\}}}}} & {{Equation}\mspace{14mu} 17}\end{matrix}$

where

X _(l) +jY _(l) =e ^(j2πl/M)   Equation 18

and j=√{square root over (−1)}. For a received sample of consecutiveMPSK symbols of length N (r₁,r₂, . . . ,r_(N)), the probability densityfunction can be expressed as:

$\begin{matrix}{{f_{N}\left( {r_{1},r_{2},\ldots \mspace{14mu},r_{N}} \right)} = {\prod\limits_{k = 1}^{N}\; {f\left( {x_{k},y_{k}} \right)}}} & {{Equation}\mspace{14mu} 19}\end{matrix}$

Using Equation 9, the partial derivative of Equation 19 with respect toamplitude A is performed and set to zero, resulting in the followingequation:

$\begin{matrix}{{\sum\limits_{k = 1}^{N}\; \frac{\frac{\partial{f\left( {x_{k},y_{k}} \right)}}{\partial A}}{f\left( {x_{k},y_{k}} \right)}} = 0} & {{Equation}\mspace{14mu} 20}\end{matrix}$

According to Equation 20, the equation for amplitude A which maximizesthe probability function is derived and expressed as follows:

$\begin{matrix}{A = {\frac{1}{N}{\sum\limits_{k = 1}^{N}\; \frac{\sum\limits_{l = 0}^{M - 1}{\left\lbrack {{x_{k}X_{l}} + {y_{k}Y_{l}}} \right\rbrack \exp \left\{ \frac{\left( {{x_{k}X_{l}} + {y_{k}Y_{l}}} \right)A}{\sigma^{2}} \right\}}}{\sum\limits_{l = 0}^{M - 1}\; {\exp \left\{ \frac{\left( {{x_{k}X_{l}} + {y_{k}Y_{l}}} \right)A}{\sigma^{2}} \right\}}}}}} & {{Equation}\mspace{14mu} 21}\end{matrix}$

If the actual SNR is high, an acceptable decision-directed amplitudeestimation is then:

$\begin{matrix}{\hat{A} \approx {\frac{1}{N}{\sum\limits_{k = 1}^{N}\; \left\lbrack {{x_{k}{\hat{X}}_{k}} + {y_{k}{\hat{Y}}_{k}}} \right\rbrack}}} & {{Equation}\mspace{14mu} 22}\end{matrix}$

where ({circumflex over (X)}_(k), Ŷ_(k)) is the estimated signal thatmaximizes X_(l) and Y_(l):

$\begin{matrix}{\left( {{\hat{X}}_{k},{\hat{Y}}_{k}} \right) = {\arg \left\{ {\max\limits_{X_{l},Y_{l}}\left\{ {{{x_{k}X_{l}} + {y_{k}Y_{l}}},{l = 0},1,\ldots \mspace{14mu},{M - 1}} \right\}} \right\}}} & {{Equation}\mspace{14mu} 23}\end{matrix}$

A method 100 for an iterative SNR estimation for a BPSK signal usingEquation 10 is shown in FIG. 1. Given an amplitude estimate A₀ and anoise variance estimate σ₀ ², a new amplitude estimate A₁ is calculatedby Equation 24, which is based on Equation 10:

$\begin{matrix}{{A_{1} = {\frac{1}{N}{\sum\limits_{k = 1}^{N}\; {r_{k}{{th}\left( \frac{A_{0}r_{k}}{\sigma_{0}^{2}} \right)}}}}},} & {{Equation}\mspace{14mu} 24}\end{matrix}$

and a new noise variance estimate σ₁ ² by:

$\begin{matrix}{\sigma_{1}^{2} = {{\frac{1}{N}{\sum\limits_{k = 1}^{N}\; r_{k}^{2}}} - A_{1}^{2}}} & {{Equation}\mspace{14mu} 25}\end{matrix}$

As the method is updated, A₀ ²/σ₀ ² converges to A₁ ²/σ₁ ². Since theSNR to be estimated is unknown, an initial SNR is assumed (step 101),denoted as:

SNR₀ =A ₀ ²/σ₀ ²   Equation 26

In step 102, corresponding values for A₀ and σ² are calculated as:

$\begin{matrix}{A_{0} = \sqrt{\frac{S\; N\; R_{0}}{1 + {S\; N\; R_{0}}}}} & {{Equation}\mspace{14mu} 27}\end{matrix}$

and

$\begin{matrix}{\sigma_{0}^{2} = {\frac{1}{1 + {S\; N\; R_{0}}}.}} & {{Equation}\mspace{14mu} 28}\end{matrix}$

Next in step 103, Equations 24 and 25 are used to calculate A₁, σ₁ ²,and SNR₁ is calculated in step 104 by Equation 29:

SNR₁ =A ₁ ²/σ₁ ²   Equation 29

Step 105 performs a decision as to whether estimate SNR₀ is within apredetermined acceptable resolution compared to the calculated SNR₁. Ifthe resolution is acceptable, then SNR₀ can be accepted as the finalestimate (step 107). Otherwise, SNR₀ is adjusted (step 106) and theprocess repeats starting at step 102. As an example with a predeterminedacceptable resolution of 0.1 dB as the benchmark, steps 102 through 106are repeated until the difference between calculated SNR₁ and estimateSNR₀ is less than or equal to 0.1 dB. Alternatively, steps 102 through106 are repeated for a predetermined number of times before bringing anend to the estimation process (step 107), and accepting the resultingestimate value, regardless of the intermediate resolutions.

A similar method for MPSK signals can be performed by replacing Equation24 in step 103 with Equation 30, which is based on Equation 21, tocalculate amplitude A₁:

$\begin{matrix}{A_{1} = {\frac{1}{N}{\sum\limits_{k = 1}^{N}\; \frac{\sum\limits_{l = 0}^{M - 1}{\left\lbrack {{x_{k}X_{l}} + {y_{k}Y_{l}}} \right\rbrack \exp \left\{ \frac{\left( {{x_{k}X_{l}} + {y_{k}Y_{l}}} \right)A_{0}}{\sigma_{0}^{2}} \right\}}}{\sum\limits_{l = 0}^{M - 1}\; {\exp \left\{ \frac{\left( {{x_{k}X_{l}} + {y_{k}Y_{l}}} \right)A_{0}}{\sigma_{0}^{2}} \right\}}}}}} & {{Equation}\mspace{14mu} 30}\end{matrix}$

FIG. 3 shows curves of calculated SNRs versus assumed SNRs for twenty1024-point sample vectors each with a real SNR of 3 dB. Each curvecrosses the straight line “calculated SNR=assumed SNR” at one point. Thecrossing point is the estimated SNR for a converged method. It isnoteworthy that the crossing points are concentrated around the true SNRof 3 dB. Variations among the 20 curves are due to the random nature ofthe noise component during each trial. The calculated values varyapproximately between −1 dB and +0.5 dB. When the assumed SNR value isgreater than the actual SNR, the calculated SNR value is less than theassumed value. This relationship is useful for quick convergence as eachsuccessive assumed SNR value can properly be increased or reducedaccordingly.

An alternative method is to iteratively solve for amplitude A, then tocompute the SNR estimate upon convergence, as shown by flow diagram ofmethod 200 in FIG. 2. In Step 201, the received vector is normalizedsuch that:

$\begin{matrix}{{\frac{1}{N}{\sum\limits_{k = 1}^{N}\; r_{k}^{2}}} = 1} & {{Equation}\mspace{14mu} 31}\end{matrix}$

Assumed minimum and maximum amplitudes of interest A_(min) and A_(max)are selected, and a predetermined resolution Δ is selected. Values A₀and A₁ are initialized as follows: A₀=A_(min) and A₁=A_(max).

In steps 202 and 203, the mean of A₀ and A₁ is calculated by:

A _(m)=(A ₀ +A ₁)/2   Equation 32

and the corresponding noise variances are determined by:

σ₀ ²=1−A ₀ ²   Equation 33

σ₁ ²=1−A ₁ ²   Equation 34

σ_(m) ²=1−A _(m) ²   Equation 35

In step 204, three estimated amplitude values, A′₀, A′₁ and A′_(m) arecalculated using Equation 24 by substituting the initial amplitudevalues A₀, A₁ and A_(m), for A₀ in Equation 24 and initial noisevariances σ₀, σ₁ and σ_(m), respectively, for σ₀ in Equation 24.

For step 205, if A_(m)>A′_(m), then the maximum amplitude A₁ is updatedas follows: A₁=A′_(m). Otherwise, the minimum amplitude A₀ is updated:A₀=A′_(m). In an alternative embodiment for step 205, if A_(m)>A′_(m),then amplitude A₁ can be updated so that A₁=A_(m); otherwise the minimumamplitude A₀ is updated: A₀=A_(m).

For step 206, the resolution Δ is evaluated. If A₁−A₀<Δ, then theestimated amplitude is the updated value A_(OUT)=(A₀+A₁)/2 with eitherA₀ or A₁ as updated amplitude values from step 205. The final estimatedsignal-to-noise ratio SNR_(OUT) is calculated from the estimatedamplitude value A_(OUT) as follows: SNR_(OUT)=A_(OUT) ²/(1−A_(OUT) ²).Otherwise the process is repeated by returning to step 202 and repeatingthe steps through step 206 until an acceptable resolution Δ is achieved.

As with method 100, method 200 can be modified to accommodate an MPSKsignal. This is achieved by calculating amplitude estimates A′₀, A′₁ andA′_(m) using Equation 30 instead of Equation 24 in step 204.

The lower bias of method 200 can be seen in FIGS. 4-13 in which the meanSNR and normalized mean square error (MSE) results are compared againstvarious SNR algorithms. Simulation results for the iterative SNRestimation method 200 are graphically compared to the RXDA, the TXDA,and the Equation 3 SNR estimation algorithms as shown in FIGS. 4-5. Asaforementioned, the TXDA algorithm is based on exact knowledge of thereceived data, which is only applicable to known training sequences. TheTXDA curve is therefore shown as a baseline for comparison purposes.

FIG. 4 shows means of the various SNR estimations generated using areceived vector of 1024 samples (N=1024) versus the actual SNR. Theiterative SNR estimation method 200 has a lower bias (excepting theknown data case) and the useful range extends down to about −5 dB. Forcomparison, the useful range in each case for RXDA and Equation 3algorithms only extends down to about 8 dB.

FIG. 5 shows the normalized mean square error (MSE) of the SNRestimations where N=1024 and also shows the Cramer-Rao (CR) bound thatis lower bounded by

${C\; R} \geq {2{\left\{ {\frac{2}{A^{2}N} + \frac{1}{N}} \right\}.}}$

The estimation by method 200 produces results having a lower normalizedMSE than that for RXDA and Equation 3.

FIGS. 6-9 show mean and MSE results of method 200 compared with RXDA,TXDA, and decision-directed for an 8PSK signal. Comparison of FIGS. 6, 7to FIGS. 8, 9 show the improvement in mean and MSE versus SNR by method200 when the sample length is increased from N=100 to N=1024,respectively. It should be noted that improvements are evident formethod 200 whereas those for Equations 1 and 3 show no improvement.

Similarly, FIGS. 10-11 show mean and MSE results for a 16PSK signal forN=100 and FIGS. 12, 13 show mean and MSE results for a 16PSK signal forN=1024 with similar results.

FIG. 14 shows several trajectories of convergence within 9 iterationsfor method 200. In general, the number of iterations depends on A_(min),A_(max) and the resolution Δ. In this example, A_(min)=0.001,A_(max)=0.999 and Δ=0.0002. As shown in FIG. 14, the estimated SNRstabilizes after 7 iterations and by the 9^(th) iteration, A₁−A₀<Δ, andthe estimation is finished.

FIG. 15 shows an embodiment for using methods 100 and 200, comprising asystem for wireless communications, such as CDMA, with base station 301and user equipments (UEs) 302-305. Base station 301 and (UEs) 302-305each include an SNR estimator which performs the low bias SNR estimationmethod 200. Improved SNR estimation provides several advantages for basestation and UE performance. For instance, improved SNR estimation for aUE is enhanced power control from having more accurate assessment of therequired uplink power. At the base station, improved channel selectionresults from better SNR estimation.

1. A method comprising: selecting a first estimate of a signal-to-noiseratio (SNR); calculating an amplitude value based on a probabilitydensity and a noise variance based on the amplitude value; calculating asecond SNR based on the amplitude value and the noise variance;calculating a resolution value based on the difference between the firstSNR estimate and the second SNR; adjusting the first SNR estimate; andperforming estimation iterations until the resolution value is equal toa predetermined value.
 2. The method as in claim 1, wherein theestimation iterations are based on the adjusted first SNR.
 3. The methodas in claim 1, further comprising performing a low-biased estimation ofa SNR of a sequence of received BPSK communication symbols.
 4. Themethod as in claim 1, further comprising performing a low-biasedestimation of a SNR of a sequence of received M-ary PSK communicationsymbols, where M represents a quantity of phases for PSK.